99 ideas
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 14713 | Truth in a scenario is the negation in that scenario being a priori incoherent [Chalmers] |
| 10098 | The 'power set' of A is all the subsets of A [George/Velleman] |
| 10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman] |
| 10101 |
Cartesian Product A x B: the set of all ordered pairs |
| 10103 | Grouping by property is common in mathematics, usually using equivalence [George/Velleman] |
| 10104 | 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman] |
| 10096 | Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman] |
| 10097 | Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman] |
| 10100 | Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman] |
| 17900 | The Axiom of Reducibility made impredicative definitions possible [George/Velleman] |
| 10109 | ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman] |
| 10108 | As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman] |
| 10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
| 18946 | Unreflectively, we all assume there are nonexistents, and we can refer to them [Reimer] |
| 10129 | A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman] |
| 10105 | Differences between isomorphic structures seem unimportant [George/Velleman] |
| 10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman] |
| 10126 | A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman] |
| 10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman] |
| 10127 | A 'complete' theory contains either any sentence or its negation [George/Velleman] |
| 10106 | Rational numbers give answers to division problems with integers [George/Velleman] |
| 10102 | The integers are answers to subtraction problems involving natural numbers [George/Velleman] |
| 10107 | Real numbers provide answers to square root problems [George/Velleman] |
| 9946 | Logicists say mathematics is applicable because it is totally general [George/Velleman] |
| 10125 | The classical mathematician believes the real numbers form an actual set [George/Velleman] |
| 17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman] |
| 10128 | The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman] |
| 17902 | A successor is the union of a set with its singleton [George/Velleman] |
| 10133 | Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman] |
| 10130 | Set theory can prove the Peano Postulates [George/Velleman] |
| 10089 | Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman] |
| 10131 | If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman] |
| 10092 | In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman] |
| 10094 | The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman] |
| 10095 | Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman] |
| 17901 | Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman] |
| 10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman] |
| 10134 | Much infinite mathematics can still be justified finitely [George/Velleman] |
| 10123 | The intuitionists are the idealists of mathematics [George/Velleman] |
| 10124 | Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman] |
| 2392 | Properties supervene if you can't have one without the other [Chalmers] |
| 2393 | Logical supervenience is when one set of properties must be accompanied by another set [Chalmers] |
| 2394 | Natural supervenience is when one set of properties is always accompanied by another set [Chalmers] |
| 2398 | Reduction requires logical supervenience [Chalmers] |
| 16048 | Physicalism says in any two physically indiscernible worlds the positive facts are the same [Chalmers, by Bennett,K] |
| 2401 | All facts are either physical, experiential, laws of nature, second-order final facts, or indexical facts about me [Chalmers] |
| 16424 | Strong metaphysical necessity allows fewer possible worlds than logical necessity [Chalmers] |
| 16425 | Metaphysical necessity is a bizarre, brute and inexplicable constraint on possibilities [Chalmers] |
| 16426 | How can we know the metaphysical impossibilities; the a posteriori only concerns this world [Chalmers] |
| 13956 | Kripke is often taken to be challenging a priori insights into necessity [Chalmers] |
| 13963 | Maybe logical possibility does imply conceivability - by an ideal mind [Chalmers] |
| 16473 | Modal Rationalism: conceivability gives a priori access to modal truths [Chalmers, by Stalnaker] |
| 19258 | Evaluate primary possibility from some world, and secondary possibility from this world [Chalmers, by Vaidya] |
| 2407 | One can wrongly imagine two things being non-identical even though they are the same (morning/evening star) [Chalmers] |
| 2390 | We attribute beliefs to people in order to explain their behaviour [Chalmers] |
| 14712 | A sentence is a priori if no possible way the world might actually be could make it false [Chalmers] |
| 2397 | 'Perception' means either an action or a mental state [Chalmers] |
| 2422 | The structure of the retina has already simplified the colour information which hits it [Chalmers] |
| 2396 | Reductive explanation is not the be-all and the end-all of explanation [Chalmers] |
| 2426 | Why are minds homogeneous and brains fine-grained? [Chalmers] |
| 2391 | Can we be aware but not conscious? [Chalmers] |
| 2412 | Can we explain behaviour without consciousness? [Chalmers] |
| 2386 | Hard Problem: why brains experience things [Chalmers] |
| 2416 | What turns awareness into consciousness? [Chalmers] |
| 2423 | Going down the scale, where would consciousness vanish? [Chalmers] |
| 2403 | Nothing in physics even suggests consciousness [Chalmers] |
| 2400 | Is intentionality just causal connections? [Chalmers] |
| 2389 | Sometimes we don't notice our pains [Chalmers] |
| 2419 | Why should qualia fade during silicon replacement? [Chalmers] |
| 2402 | It seems possible to invert qualia [Chalmers] |
| 2415 | In blindsight both qualia and intentionality are missing [Chalmers] |
| 2414 | When distracted we can totally misjudge our own experiences [Chalmers] |
| 2409 | Maybe dualist interaction is possible at the quantum level? [Chalmers] |
| 2411 | Supervenience makes interaction laws possible [Chalmers] |
| 2424 | It is odd if experience is a very recent development [Chalmers] |
| 2413 | If I can have a zombie twin, my own behaviour doesn't need consciousness [Chalmers] |
| 2417 | Does consciousness arise from fine-grained non-reductive functional organisation? [Chalmers] |
| 2428 | Maybe the whole Chinese Room understands Chinese, though the person doesn't [Chalmers] |
| 2418 | The Chinese Mind doesn't seem conscious, but then nor do brains from outside [Chalmers] |
| 2406 | H2O causes liquidity, but no one is a dualist about that [Chalmers] |
| 2405 | Perhaps consciousness is physically based, but not logically required by that base [Chalmers] |
| 2395 | Zombies imply natural but not logical supervenience [Chalmers] |
| 9318 | Phenomenal consciousness is fundamental, with no possible nonphenomenal explanation [Chalmers, by Kriegel/Williford] |
| 2404 | Nothing external shows whether a mouse is conscious [Chalmers] |
| 2429 | Temperature (etc.) is agreed to be reducible, but it is multiply realisable [Chalmers] |
| 18403 | Indexicals may not be objective, but they are a fact about the world as I see it [Chalmers] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 14708 | Rationalist 2D semantics posits necessary relations between meaning, apriority, and possibility [Chalmers, by Schroeter] |
| 13958 | The 'primary intension' is non-empirical, and fixes extensions based on the actual-world reference [Chalmers] |
| 2399 | Meaning has split into primary ("watery stuff"), and secondary counterfactual meaning ("H2O") [Chalmers] |
| 13959 | The 'secondary intension' is determined by rigidifying (as H2O) the 'water' picked out in the actual world [Chalmers] |
| 13957 | Primary and secondary intensions are the a priori (actual) and a posteriori (counterfactual) aspects of meaning [Chalmers] |
| 13961 | We have 'primary' truth-conditions for the actual world, and derived 'secondary' ones for counterfactual worlds [Chalmers] |
| 14739 | 'Water' is two-dimensionally inconstant, with different intensions in different worlds [Chalmers, by Sider] |
| 13962 | Two-dimensional semantics gives a 'primary' and 'secondary' proposition for each statement [Chalmers] |
| 13960 | In two-dimensional semantics we have two aspects to truth in virtue of meaning [Chalmers] |
| 16427 | Presumably God can do anything which is logically possible [Chalmers] |